Optimal entry and exit for variance swaps: closed-form rules for the perpetual contract

Abstract

Variance swaps are a convenient instrument for trading vega and convexity, and a listed contract now trades on Cboe. We ask when a trader should put such a position on and when she should take it off, and for a perpetual, continuously settled contract we answer both in closed form: each threshold is the unique root of a smooth-pasting equation in confluent hypergeometric functions. Under the pricing measure the question has no content, the mark-to-market being a martingale. Under the physical measure with a variance risk premium it becomes meaningful, and then reduces: the accrued variance separates exactly, the maturity, strike and costs are absorbed into a single forcing term whose sign fixes the geometry of the exercise region, and what is left on the perpetual is an affine reward on a CIR process, which the optimal-stopping literature already solves. Entering the position and exiting it are not mirror images. An exit rule follows from the premium and the trading spread, both observable. For the short --- the only side worth opening under the empirical sign of the premium --- an entry rule exists only for an interval of carrying charges, and even there triggers only deep in the upper tail of the physical law: a trader who is out of the market pays nothing to stay out, so an operational entry rule needs a cost of idle capital that the exit rule does not.